Algebraic Expressions

Mathematics J.S.S 2 Second Term

Theme: Algebraic Processes

WEEK 2

Algebraic Expressions

Performance Objectives

Student should be able to:

1. Apply use of quadratic equation box in expanding and factorizing algebraic expressions.
2. Solve quantitative reasoning problem

Expansions Leading to Quadratic Expressions

Content

A quadratic expression is one in which 2 is the highest power of the unknown(s) in the expression. For example, x² – 4x – 12, 16 – a², 3x² + 17xy + 10y² are all quadratic expressions.

Example:

Expand: (a) (a + 3) (a – 4)

    (b) (2x + 3) (4x – 5)

Solution:

1.    (a + 3) (a – 4)

      =a (a – 4) + 3(a – 4)

      = a² – 4a + 3a – 12

      = a² – a – 12

2. (2x + 3) (4x – 5)

= 4x (2x + 3) – 5(2x + 3)

= 8x² + 12x – 10x – 15

= 8x² + 2x – 15

  Apart from this method Quadratic Expansion to form Quadratic Expression using Box Method.\

6. Now sum up the one in the middle i.e the like terms 3X² - 4X + 6X - 8

= 3X² + 2X – 8  (and this make it to form quadratic expression)

Example 2

Expand (t + 5)(t – 2)

6. Now sum up the one in the middle i.e the like terms t² - 2t + 5t – 10

= t² + 3t – 10  (and this make it to form quadratic expression)

Factorization of Quadratic Expressions

A factor is a number or quantity that when multiplied with another number produces a given number or expression. For example, the factors of 20 are 1, 2, 4, 5, 10, and 20.

As we know that;

20 = 1 x 20

20 = 2 x 10

20 = 4 x 5

You can also factorize quadratic expressions using block form. Remember that factorizing an expression simplifies it in some way. Factorizing is the reverse of expanding brackets.

Example

Solve the quadratic expression 4X² – 4X – 24 using Factorization method

Solution

You solve using the following methods

1. Equate the equation to zero.

4X² – 4X - 24 = 0

2. Check if there is no greatest common factor (and the greatest common factor here is 4 i.e 4(X² – X – 6) = 0

Now this is what we are working with (X² – X – 6)

Example 2

Solve the quadratic expression X² + 2X - 24 using Factorization method

Solution

You solve using the following methods

1. Equate the equation to zero.

X² + 2X - 24 = 0

2. Check if there is no greatest common factor (and the greatest common factor here is 1 i.e 1(X² + 2X - 24) = 0

Now this is what we are working with (X² + 2X - 24)

8. Now from the factors above look for the numbers that when it is multiplied together it will give us -24 and when added together it will give us +2

Note since only – X + = - 

        + X - = -

                               - X - = +

                           + X + = +   

And the only numbers that can fit in is -4 and +6 i.e -4 X +6 = -24 & -4 + (+6) = +2

 

        X - 4 = 0                           or             X + 6 = 0

       then add  4 to both sides              then subtract 6 from both sides

 X - 4 + 4 = 0 + 4                                X + 6 - 6 = 0 - 6

       So X = +4                          or             X = -6

Therefore X = +4 or -6

Check

   X² + 2X – 24 = 0

= X² - 4X + 6X – 24 = 0

= (X² - 4X) (+6X – 24) = 0

= X(X – 4) + 6(X – 4) = 0

= (x – 4)(X + 6) = 0

   X – 4 = 0 or X + 6 = 0

   X = 4 or X = -6

Note; Always your answers after solving each question